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Time-domain decomposition for optimal control problems governed by semilinear hyperbolic systems with mixed two-point boundary conditions

Krug Richard, Leugering Günter, Martin Alexander, Schmidt Martin, Weninger Dieter

Control and Cybernetics

2021

Vol. 50, No. 4

427--455

In this article, we study the time-domain decomposition of optimal control problems for systems of semilinear hyperbolic equations and provide an in-depth well-posedness analysis. This is a continuation of our work, Krug et al. (2021) in that we now consider mixed two-point boundary value problems. The more general boundary conditions significantly enlarge the scope of applications, e.g., to hyperbolic problems on metric graphs with cycles. We design an iterative method based on the optimality systems that can be interpreted as a decomposition method for the original optimal control problem into virtual control problems on smaller time domains.

Variation of constants formula for hyperbolic systems

Lichtner M.

Journal of Applied Analysis

2009

Vol. 15, nr 1

79-100

A smooth variation of constants formula for semilinear hyperbolic systems is established using a suitable Banach space X of continuous functions together with its sun dual space [wzór]. It is shown that mild solutions of this variation of constants formula generate a smooth semiflow in X. This proves that the stability of stationary states for the nonlinear flow is determined by the stability of the linearized semigroup.

An output controllability problem for semilinear distributed hyperbolic systems

Zerrik E., Larhrissi R., Bourray H.

International Journal of Applied Mathematics and Computer Science

2007

Vol. 17, no 4

437-446

The paper aims at extending the notion of regional controllability developed for linear systems to the semilinear hyperbolic case. We begin with an asymptotically linear system and the approach is based on an extension of the Hilbert uniqueness method and Schauder’s fixed point theorem. The analytical case is then tackled using generalized inverse techniques and converted to a fixed point problem leading to an algorithm which is successfully implemented numerically and illustrated with examples.